Separation Of Variables Approach Research Articles

Small-signal transmittances of the non-ideal flyback switch-mode DC–DC converter

Small-signal transmittances of the power stage of a flyback converter in continuous conduction mode are derived on the averaged model obtained by the separation of variables approach. The precise knowledge of these transmittances is necessary in the design process of the converter control circuit. Apart from mathematical formulas for transmittances, the numerical calculations of the frequency dependencies of the transmittances for the assumed set of the converter parameters are presented with the parasitic resistances of components taken into account. The results of the calculations are compared with the measurements performed on the laboratory model of the converter and a good consistency is observed. It is concluded, that the results of the paper may be useful in the designing process of a control circuit of the flyback converter.

Wave damping by breakwater and mangrove for protecting shoreline

In this paper, we investigate wave damping caused by breakwater and mangrove using Shallow Water Equations. To derive the wave transmission coefficient, we solve the model analytically using the separation of variables approach. To solve the problem numerically, we use a staggered finite volume approach. We validate our numerical result by comparing it to the analytical solution. Furthermore, we investigate the effects of the dimension and characteristics of the breakwater and mangrove in reducing incoming wave height. We believe that this finding will aid engineers in designing the coastal protector.

Mathematical Modeling and Performance Evaluation of 3D Ferroelectric Negative Capacitance FinFET

Ferroelectric negative capacitance materials have now been proposed for lowering electronics energy dissipation beyond basic limitations. In this paper, we presented the analysis on the performance of negative capacitance (NC) FinFET in comparison with conventional gate dielectrics by using a separation of variables approach, which is an optimal quasi-3D mathematical model. The result has been signified steeper surface potential (ψ), lower threshold voltage (Vth), 1.2 mA of on-state current (Ion), and enhanced immunity of negative capacitance FinFET against short channel effects (SCE’s) like 35.3 mV/V of drain-induced barrier lowering (DIBL), 60 mV/dec of subthreshold swing (SS) along with smallest off state current (Ioff) among another conventional gate dielectric. Hence, NC FinFET can be a potential candidate for low power and high-performance device.

Input Small-Signal Characteristics of Selected DC–DC Switching Converters

The main goal of this study was to derive small-signal models of the input characteristics of buck, boost, and flyback converters working in continuous conduction mode (CCM) and discontinuous conduction mode (DCM). The models presented in the paper were derived using the separation of variables approach and included the parasitic resistances of all converter components. The paper features a discussion about the limitations of the model accuracy. The presented characteristics were obtained by calculation and verified by measurements. The input characteristics of converters are essential in the design of converters used in Power Factor Correction systems as well as in maximum power point tracking systems (MPPT).

Separation of variables in AdS/CFT: functional approach for the fishnet CFT

The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV) remains unknown in mathcal{N} = 4 SYM and similar models, even though it is widely believed they are integrable. In this paper we initiate the SoV approach for observables with nontrivial coupling dependence in a close cousin of mathcal{N} = 4 SYM — the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nontrivial observables in a form suitable for numerical evaluation. We present some applications of these methods. In particular, we discuss the possible SoV structure of the one-point correlators in presence of a defect, and write down a SoV-type expression for diagonal OPE coefficients involving an arbitrary state and the Lagrangian density operator. We believe that many of the findings of this paper can be applied in the mathcal{N} = 4 SYM case, as we speculate in the last part of the article.

On separation of variables for reflection algebras

We implement our new separation of variables (SoV) approach for open quantum integrable models associated to higher rank representations of the reflection algebras. We construct the (SoV) basis for the fundamental representations of the Y(gln) reflection algebra associated to general integrable boundary conditions. Moreover, we give the conditions on the boundary K-matrices allowing for the transfer matrix to be diagonalizable with simple spectrum. These SoV basis are then used to completely characterize the transfer matrix spectrum for the rank one and two reflection algebras. The rank one case is developed for both the rational and trigonometric fundamental representations of the 6-vertex reflection algebra. On the one hand, we extend the complete spectrum characterization to representations previously lying outside the SoV approach, e.g. those for which the standard algebraic Bethe Ansatz applies. On the other hand, we show that our new SoV construction can be reduced to the generalized Sklyanin’s one whenever it is applicable. The rank two case is developed explicitly for the fundamental representations of the Y(gl3) reflection algebra associated to general integrable boundary conditions. For both rank one and two our SoV approach leads to a complete characterization of the transfer matrix spectrum in terms of a set of polynomial solutions to the corresponding quantum spectral curve equation. Those are finite difference functional equations of order equal to the rank plus one, i.e. here two and three respectively for the Y(gl2) and Y(gl3) reflection algebras.

Complete spectrum of quantum integrable lattice models associated to by separation of variables

In this paper we apply our new separation of variables approach to completely characterize the transfer matrix spectrum for quantum integrable lattice models associated to fundamental evaluation representations of with quasi-periodic boundary conditions. We consider here the case of generic deformations associated to a parameter q which is not a root of unity. The separation of variables (SoV) basis for the transfer matrix spectral problem is generated by using the action of the transfer matrix itself on a generic co-vector of the Hilbert space, following the general procedure described in our paper (Maillet and Niccoli 2018 J. Math. Phys. 59 091417). Such a SoV construction allows to prove that for general values of the parameters defining the model the transfer matrix is diagonalizable and with simple spectrum for any twist matrix which is also diagonalizable with simple spectrum. Then, using together the knowledge of such a SoV basis and of the fusion relations satisfied by the hierarchy of transfer matrices, we derive a complete characterization of the transfer matrix eigenvalues and eigenvectors as solutions of a system of polynomial equations of order n + 1. Moreover, we show that such a SoV discrete spectrum characterization is equivalently reformulated in terms of a finite difference functional equation, the quantum spectral curve equation, under a proper choice of the set of its solutions. A construction of the associated Q-operator induced by our SoV approach is also presented.

Heat Source Modeling in Selective Laser Melting.

Selective laser melting (SLM) is an emerging additive manufacturing (AM) technology for metals. Intricate three-dimensional parts can be generated from the powder bed by selectively melting the desired location of the powders. The process is repeated for each layer until the part is built. The necessary heat is provided by a laser. Temperature magnitude and history during SLM directly determine the molten pool dimensions, thermal stress, residual stress, balling effect, and dimensional accuracy. Laser-matter interaction is a crucial physical phenomenon in the SLM process. In this paper, five different heat source models are introduced to predict the three-dimensional temperature field analytically. These models are known as steady state moving point heat source, transient moving point heat source, semi-elliptical moving heat source, double elliptical moving heat source, and uniform moving heat source. The analytical temperature model for all of the heat source models is solved using three-dimensional differential equations of heat conduction with different approaches. The steady state and transient moving heat source are solved using a separation of variables approach. However, the rest of the models are solved by employing Green’s functions. Due to the high temperature in the presence of the laser, the temperature gradient is usually high which has a substantial impact on thermal material properties. Consequently, the temperature field is predicted by considering the temperature sensitivity thermal material properties. Moreover, due to the repeated heating and cooling, the part usually undergoes several melting and solidification cycles, and this physical phenomenon is considered by modifying the heat capacity using latent heat of melting. Furthermore, the multi-layer aspect of the metal AM process is considered by incorporating the temperature history from the previous layer since the interaction of the layers have an impact on heat transfer mechanisms. The proposed temperature field models based on different heat source approaches are validated using experimental measurement of melt pool geometry from independent experimentations. A detailed explanation of the comparison of models is also provided. Moreover, the effect of process parameters on the balling effect is also discussed.

Exact quantization conditions for the elliptic Ruijsenaars-Schneider model

We propose and test exact quantization conditions for the N-particle quantum elliptic Ruijsenaars-Schneider integrable system, as well as its Calogero-Moser limit, based on the conjectural correspondence to the five-dimensional mathcal{N}={1}^{ast } SU(N) gauge theory in the Nekrasov-Shatashvili limit. We discuss two natural sets of quantization conditions, related by the electro-magnetic duality, and the importance of non-perturbative corrections in the Planck constant. We also comment on the eigenfunction problem, by reinterpreting the Separation of Variables approach in gauge theory terms.

The open XXZ spin chain in the SoV framework: scalar product of separate states

In our previous paper (Kitanine et al 2017 J. Phys. A: Math. Theor. 50 224001) we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. We first revisit the SoV solution of this model using a matrix version of the Vertex-IRF gauge transformation so as to reduce one of the boundary K-matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to—although more cumbersome than—the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model.

ANALISIS KONVERGENSI METODE BEDA HINGGA DALAM MENGHAMPIRI PERSAMAAN DIFUSI

The diffusion equation or known as heat equation is a parabolic and linear type of partial differential equation. One of the numerical method to approximate the solution of diffusion equations is Finite Difference Method (FDM). In this study, the analysis of numerical convergence of FDM to the solution of diffusion equation is discussed. The analytical solution of diffusion equation is given by the separation of variables approach. Here, the result show the convergence of rate the numerical method is approximately approach 2. This result is in a good agreement with the spatial error from Taylor expansion of spatial second derivative.

Antiperiodic dynamical 6-vertex model by separation of variables II: functional equations and form factors

We pursue our study of the antiperiodic dynamical 6-vertex model using Sklyanin’s separation of variables approach, allowing in the model new possible global shifts of the dynamical parameter. We show in particular that the spectrum and eigenstates of the antiperiodic transfer matrix are completely characterized by a system of discrete equations. We prove the existence of different reformulations of this characterization in terms of functional equations of Baxter’s type. We notably consider the homogeneous functional T–Q equation which is the continuous analog of the aforementioned discrete system and show, in the case of a model with an even number of sites, that the complete spectrum and eigenstates of the antiperiodic transfer matrix can equivalently be described in terms of a particular class of its Q-solutions, hence leading to a complete system of Bethe equations. Finally, we compute the form factors of local operators for which we obtain determinant representations in finite volume.

The eight-vertex model with quasi-periodic boundary conditions

We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T–Q functional equations, where the Q-functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.

Comparison of transient states in step-down power converter (BUCK) in continuous and discontinuous conduction mode

Abstract The object of this paper is a step-down (BUCK) power converter working in the continuous conduction mode (CCM) or discontinuous conduction mode (DCM). Two types of transient states in a converter have been analyzed and observed experimentally: slow transients, described by averaged models of a converter and fast transients, in the course of a single switching period. The averaged models of converter working in CCM, presented in various sources are similar, with some differences concerning only the description of parasitic effects. The averaged models for DCM depend on the adopted modeling method. Models obtained by the switch averaging approach are second-order models (containing two reactive elements in equivalent circuit representation). Models obtained by the separation of variables approach are first-order models. The experimental results given in this paper show the first-order type of transients. Another group of experiments concern fast transients in the course of a single switching period. The oscillations of inductor voltage in the part of a switching period are observed for DCM.

Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model

This paper deals with the free vibration analysis of circular alumina (Al2O3) nanobeams in the presence of surface and thermal effects resting on a Pasternak foundation. The system of motion equations is derived using Hamilton’s principle under the assumptions of the classical Timoshenko beam theory. The effects of the transverse shear deformation and rotary inertia are also considered within the framework of the mentioned theory. The separation of variables approach is employed to discretize the governing equations which are then solved by an analytical method to obtain the natural frequencies of the alumina nanobeams. The results show that the surface effects lead to an increase in the natural frequency of nanobeams as compared with the classical Timoshenko beam model. In addition, for nanobeams with large diameters, the surface effects may increase the natural frequencies by increasing the thermal effects. Moreover, with regard to the Pasternak elastic foundation, the natural frequencies are increased slightly. The results of the present model are compared with the literature, showing that the present model can capture correctly the surface effects in thermal vibration of nanobeams.

SOV approach for integrable quantum models associated with general representations on spin-1/2 chains of the 8-vertex reflection algebra

The analysis of the transfer matrices associated with the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method, which generalizes to these integrable quantum models the method first introduced by Sklyanin. For representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras, for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which particularly apply to transfer matrix eigenstates. A comparison of our analysis of the 8-vertex reflection algebra with that of (Niccoli G 2012 J. Stat. Mech. P10025, Faldella S et al 2014 J. Stat. Mech. P01011) for the 6-vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products, which exists for these two different realizations of the reflection algebra once they are described by the SOV method. As will be shown in a future publication, this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8-vertex and 6-vertex type.

Equivalent circuits for averaged description of DC-DC switch-mode power converters based on separation of variables approach

Abstract Large-signal and small-signal averaged models of basic switch-mode DC-DC power converters: BUCK (step-down) and BOOST (step-up) are presented. Models are derived with the separation of variables approach and have the form of equivalent circuits, suitable for a circuit simulation. Apart from equivalent circuits, small-signal transmittances of converters for CCM and DCM modes are discussed. Parasitic resistances of all components of converters are taken into account. A few examples of simulations and measurement results of selected converter characteristics are also presented. It is shown, that neglecting parasitic resistances (often met in works of other authors) may lead to serious errors in an averaged description of converters.

Simulation of diffuse photon migration in tissue by a Monte Carlo method derived from the optical scattering of spheroids

A Monte Carlo method was derived from the optical scattering properties of spheroidal particles and used for modeling diffuse photon migration in biological tissue. The spheroidal scattering solution used a separation of variables approach and numerical calculation of the light intensity as a function of the scattering angle. A Monte Carlo algorithm was then developed which utilized the scattering solution to determine successive photon trajectories in a three-dimensional simulation of optical diffusion and resultant scattering intensities in virtual tissue. Monte Carlo simulations using isotropic randomization, Henyey-Greenstein phase functions, and spherical Mie scattering were additionally developed and used for comparison to the spheroidal method. Intensity profiles extracted from diffusion simulations showed that the four models differed significantly. The depth of scattering extinction varied widely among the four models, with the isotropic, spherical, spheroidal, and phase function models displaying total extinction at depths of 3.62, 2.83, 3.28, and 1.95 cm, respectively. The results suggest that advanced scattering simulations could be used as a diagnostic tool by distinguishing specific cellular structures in the diffused signal. For example, simulations could be used to detect large concentrations of deformed cell nuclei indicative of early stage cancer. The presented technique is proposed to be a more physical description of photon migration than existing phase function methods. This is attributed to the spheroidal structure of highly scattering mitochondria and elongation of the cell nucleus, which occurs in the initial phases of certain cancers. The potential applications of the model and its importance to diffusive imaging techniques are discussed.

Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables

The antiperiodic transfer matrices associated to higher spin representations of the rational 6-vertex Yang-Baxter algebra are analyzed by generalizing the approach introduced recently in the framework of Sklyanin's quantum separation of variables (SOV) for cyclic representations, spin-1/2 highest weight representations, and also for spin-1/2 representations of the 6-vertex reflection algebra. Such SOV approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on Bethe ansatz and Baxter Q-operator. In particular, we both prove the completeness of the SOV characterization of the transfer matrix spectrum and its simplicity. Then, the derived characterization of local operators by Sklyanin's quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products.

Correctness Of Free Form Surface FittingAlgorithms

In this paper we consider the problem of fitting a fixed, free form surface to coordinate data. We describe an algorithm using a separation of variables approach that allows probe radius compensation to be implemented in an accurate and simple way. We also show how to generate test data with which to validate surface,fitting algorithms. By comparison with the surface3itting problem, the data generation is extremely straightforward and requires onlra capability to generate points lying exactly on a surface along with their corresponding normal vectors. The data generation also caters for probe radius compensation in a simple way. The main advantage of the datageneration scheme is that it allows the behaviour of approximate solution algorithms to be assessed without the need to implement a comprehensive algorith~n that addresses the true computational aim. We illustrate this approach on aspheric surfaces.